Optimization Modeling

Chapter 4
Sensitivity Analysis
This chapter
The subject of this chapter is the introduction of marginal values (shadow
prices and reduced costs) and sensitivity ranges which are tools used when
conducting a sensitivity analysis of a linear programming model. A sensitivity
analysis investigates the changes to the optimal solution of a model as the
result of changes in input data.
References
Sensitivity analysis is discussed in a variety of text books. A basic treatment
can be found in, for instance, [Ch83], [Ep87] and [Ko87].
4.1 Introduction
Terminology
In a linear program, slack variables may be introduced to transform an inequality
constraint into an equality constraint. When the simplex method is used to
solve a linear program, it calculates an optimal solution (i.e. optimal values for
the decision and/or slack variables), an optimal objective function value, and
partitions the variables into basic variables and nonbasic variables. Nonbasic
variables are always at one of their bounds (upper or lower), while basic variables
are between their bounds. The set of basic variables is usually referred
to as the optimal basis and the corresponding solution is referred to as the
basic solution. Whenever one or more of the basic variables (decision and/or
slack variables) happen to be at one of their bounds, the corresponding basic
solution is said to be degenerate.
Marginal values
The simplex algorithm gives extra information in addition to the optimal solution.
The algorithm provides marginal values which give information on the
variability of the optimal solution to changes in the data. The marginal values
are divided into two groups:
shadow prices which are associated with constraints and their right-hand
side, and
reduced costs which are associated with the decision variables and their
bounds.
These are discussed in the next two sections.